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Properties

Label 20.0.11890568922...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{28}\cdot 3^{18}\cdot 5^{15}\cdot 7^{17}\cdot 11^{5}$
Root discriminant $225.82$
Ramified primes $2, 3, 5, 7, 11$
Class number $40$ (GRH)
Class group $[2, 20]$ (GRH)
Galois group $D_4\times F_5$ (as 20T42)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1918665666, -5069452068, 6350854338, -4763924676, 1833189930, 280326630, -380644878, -146915028, 46232796, 65471640, 2579485, -2263018, 1040871, 118446, -19644, 10716, -312, -264, 75, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 75*x^18 - 264*x^17 - 312*x^16 + 10716*x^15 - 19644*x^14 + 118446*x^13 + 1040871*x^12 - 2263018*x^11 + 2579485*x^10 + 65471640*x^9 + 46232796*x^8 - 146915028*x^7 - 380644878*x^6 + 280326630*x^5 + 1833189930*x^4 - 4763924676*x^3 + 6350854338*x^2 - 5069452068*x + 1918665666)
 
gp: K = bnfinit(x^20 - 10*x^19 + 75*x^18 - 264*x^17 - 312*x^16 + 10716*x^15 - 19644*x^14 + 118446*x^13 + 1040871*x^12 - 2263018*x^11 + 2579485*x^10 + 65471640*x^9 + 46232796*x^8 - 146915028*x^7 - 380644878*x^6 + 280326630*x^5 + 1833189930*x^4 - 4763924676*x^3 + 6350854338*x^2 - 5069452068*x + 1918665666, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 75 x^{18} - 264 x^{17} - 312 x^{16} + 10716 x^{15} - 19644 x^{14} + 118446 x^{13} + 1040871 x^{12} - 2263018 x^{11} + 2579485 x^{10} + 65471640 x^{9} + 46232796 x^{8} - 146915028 x^{7} - 380644878 x^{6} + 280326630 x^{5} + 1833189930 x^{4} - 4763924676 x^{3} + 6350854338 x^{2} - 5069452068 x + 1918665666 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(118905689229936268771490188271616000000000000000=2^{28}\cdot 3^{18}\cdot 5^{15}\cdot 7^{17}\cdot 11^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $225.82$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 3, 5, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{10}$, $\frac{1}{6} a^{15} - \frac{1}{6} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5}$, $\frac{1}{66} a^{16} + \frac{1}{66} a^{15} + \frac{2}{33} a^{14} + \frac{1}{11} a^{13} - \frac{1}{2} a^{12} - \frac{5}{22} a^{11} - \frac{2}{11} a^{10} + \frac{5}{11} a^{9} + \frac{2}{11} a^{8} + \frac{4}{33} a^{7} - \frac{1}{3} a^{6} - \frac{5}{33} a^{5} + \frac{3}{11} a^{4} - \frac{3}{11} a^{3} - \frac{1}{11} a - \frac{5}{11}$, $\frac{1}{132} a^{17} - \frac{1}{132} a^{16} - \frac{3}{44} a^{15} + \frac{3}{44} a^{14} + \frac{7}{44} a^{13} + \frac{17}{44} a^{12} + \frac{17}{44} a^{11} - \frac{15}{44} a^{10} + \frac{3}{22} a^{9} - \frac{4}{33} a^{8} + \frac{7}{33} a^{7} - \frac{9}{22} a^{6} - \frac{1}{22} a^{5} + \frac{1}{11} a^{4} - \frac{5}{22} a^{3} - \frac{1}{22} a^{2} - \frac{3}{22} a - \frac{1}{22}$, $\frac{1}{25212} a^{18} + \frac{79}{25212} a^{17} - \frac{1}{132} a^{16} + \frac{2003}{25212} a^{15} + \frac{4909}{25212} a^{14} - \frac{2575}{8404} a^{13} - \frac{625}{8404} a^{12} + \frac{51}{8404} a^{11} - \frac{1823}{4202} a^{10} - \frac{2257}{6303} a^{9} - \frac{2731}{6303} a^{8} + \frac{1015}{12606} a^{7} + \frac{4679}{12606} a^{6} - \frac{1270}{6303} a^{5} - \frac{415}{4202} a^{4} - \frac{73}{4202} a^{3} + \frac{1699}{4202} a^{2} - \frac{2053}{4202} a + \frac{875}{2101}$, $\frac{1}{136665241911260358177209604117655760134197883518428879455037374626705042731855677626778529388} a^{19} + \frac{1235332555847494883823553909281109925733067481069929789933133948315959959799757869351859}{136665241911260358177209604117655760134197883518428879455037374626705042731855677626778529388} a^{18} - \frac{66969447304980197606793882938657917814468234228030534811177067114849884427744707534417986}{34166310477815089544302401029413940033549470879607219863759343656676260682963919406694632347} a^{17} + \frac{488991462958870883256741742528216929791452423541704702169655484412741270555011463576908667}{68332620955630179088604802058827880067098941759214439727518687313352521365927838813389264694} a^{16} - \frac{969797719816178210047204401123231382137457219006509588866425196874662209437615784956065461}{22777540318543393029534934019609293355699647253071479909172895771117507121975946271129754898} a^{15} + \frac{8014156270044952987425423576842581406753656634407348435961166239387179517611251156419864572}{34166310477815089544302401029413940033549470879607219863759343656676260682963919406694632347} a^{14} - \frac{3569991357679596662515065678656321829530178304742518276044237072765208146158968133655692627}{11388770159271696514767467009804646677849823626535739954586447885558753560987973135564877449} a^{13} - \frac{124849853642401944655561580794892886416949503482721427773943271378960563618621301877799356}{11388770159271696514767467009804646677849823626535739954586447885558753560987973135564877449} a^{12} + \frac{9645608633724052185449380372586973139056922597825473400752622709228720954587809176560162385}{45555080637086786059069868039218586711399294506142959818345791542235014243951892542259509796} a^{11} - \frac{31547333266449616989410322373025961170542469016604841372623480400536840006726220088329029549}{136665241911260358177209604117655760134197883518428879455037374626705042731855677626778529388} a^{10} - \frac{3061432641829145205204253267204484016342720222342323604686580617204651060688358828751105393}{6212056450511834462600436550802534551554449250837676338865335210304774669629803528489933154} a^{9} - \frac{23002765002166763165577603759641524503144684456979087273931799600566881702380921738024022343}{68332620955630179088604802058827880067098941759214439727518687313352521365927838813389264694} a^{8} - \frac{32924607512636994119707430974859831080015830452763014817735451347375640520301660695469174881}{68332620955630179088604802058827880067098941759214439727518687313352521365927838813389264694} a^{7} - \frac{11061456354334913759781672780903448196385188565858908550393909691293897371511906297592904683}{22777540318543393029534934019609293355699647253071479909172895771117507121975946271129754898} a^{6} - \frac{12701402427515199436488882933868464043820593397563334080882748384959425584459884420306763530}{34166310477815089544302401029413940033549470879607219863759343656676260682963919406694632347} a^{5} + \frac{8030721978727568355922790961953080088720595551357508759217369009744165019612353778116909579}{22777540318543393029534934019609293355699647253071479909172895771117507121975946271129754898} a^{4} - \frac{775786780962653431572351530623230273881649462610639870714624813502158825869487658381120662}{11388770159271696514767467009804646677849823626535739954586447885558753560987973135564877449} a^{3} + \frac{6431616916573121868497621485019848731765551234580112371211599129653746615531384467020387}{11388770159271696514767467009804646677849823626535739954586447885558753560987973135564877449} a^{2} - \frac{568374302553019752773404600062263662816066246248285177976947748243403404130931120019792491}{2070685483503944820866812183600844850518149750279225446288445070101591556543267842829977718} a - \frac{4469919139318874748297089652770732742665258352287803716105469378343718562156775233723648145}{22777540318543393029534934019609293355699647253071479909172895771117507121975946271129754898}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{20}$, which has order $40$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 742303713446840.1 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4\times F_5$ (as 20T42):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 160
The 25 conjugacy class representatives for $D_4\times F_5$
Character table for $D_4\times F_5$ is not computed

Intermediate fields

\(\Q(\sqrt{15}) \), 4.0.1386000.2, 5.1.388962000.4, 10.2.145239779946240000000.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 siblings: data not computed
Degree 40 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R R R R ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.10.14.1$x^{10} - 2 x^{6} + 2 x^{5} + 2 x^{2} + 2$$10$$1$$14$$F_{5}\times C_2$$[2]_{5}^{4}$
2.10.14.1$x^{10} - 2 x^{6} + 2 x^{5} + 2 x^{2} + 2$$10$$1$$14$$F_{5}\times C_2$$[2]_{5}^{4}$
$3$3.10.9.2$x^{10} + 3$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
3.10.9.2$x^{10} + 3$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
5Data not computed
$7$7.10.8.1$x^{10} - 7 x^{5} + 147$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
7.10.9.1$x^{10} - 7$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
$11$11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$